This viewpoint particularly fits with (the aim of) the present paper: we authors never worked specifically on billiards, but rather we realized that certain (fairly recent) theorems of arithmetico-geometrical nature may be applied so to yield, after suitable work, natural conclusions in the realm of elliptical billiards. More precisely, these results have implications regarding periodical orbits and finiteness of billiard trajectories with certain (simple and seemingly natural) patterns or properties which we shall introduce. Naturally the study of elliptical billiards goes back to long ago P. Sarnak pointed out to us the reference [1]} by the Jesuit priest R. Boscovich, an astronomer and scientist, going back to the XVIII Centrury. and is present also in recent literature (see e.g. the book [2]}, the paper [3]} by A. Connes and D. Zagier, and the paper [4]} by D. Reznik, R. Garcia, J. Koiller); however apparently no situation similar to what appears in this paper has been already analysed.
